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Limit theorems for nearly unstable Hawkes processes. (English) Zbl 1319.60101
Summary: Because of their tractability and their natural interpretations in term of market quantities, Hawkes processes are nowadays widely used in high-frequency finance. However, in practice, the statistical estimation results seem to show that very often, only nearly unstable Hawkes processes are able to fit the data properly. By nearly unstable, we mean that the \(L^{1}\) norm of their kernel is close to unity. We study in this work such processes for which the stability condition is almost violated. Our main result states that after suitable rescaling, they asymptotically behave like integrated Cox-Ingersoll-Ross models. Thus, modeling financial order flows as nearly unstable Hawkes processes may be a good way to reproduce both their high and low frequency stylized facts. We then extend this result to the Hawkes-based price model introduced by E. Bacry et al. [Quant. Finance 13, No. 1, 65–77 (2013; Zbl 1280.91073)]. We show that under a similar criticality condition, this process converges to a Heston model. Again, we recover well-known stylized facts of prices, both at the microstructure level and at the macroscopic scale.

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
91B25 Asset pricing models (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
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[1] Adamopoulos, L. (1976). Cluster models for earthquakes: Regional comparisons. Journal of the International Association for Mathematical Geology 8 463-475.
[2] Aït-Sahalia, Y., Cacho-Diaz, J. and Laeven, R. J. (2010). Modeling financial contagion using mutually exciting jump processes. Technical report, National Bureau of Economic Research, Cambridge, MA.
[3] Alaya, M. B. and Kebaier, A. (2012). Parameter estimation for the square-root diffusions: Ergodic and nonergodic cases. Stoch. Models 28 609-634. · Zbl 06117449
[4] Bacry, E., Dayri, K. and Muzy, J.-F. (2012). Non-parametric kernel estimation for symmetric Hawkes processes. Application to high frequency financial data. Eur. Phys. J. B 85 1-12.
[5] Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J. F. (2012). Scaling limits for Hawkes processes and application to financial statistics. Preprint. Available at . arXiv:1202.0842 · Zbl 1292.60032
[6] Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J. F. (2013). Modelling microstructure noise with mutually exciting point processes. Quant. Finance 13 65-77. · Zbl 1280.91073
[7] Bacry, E. and Muzy, J.-F. (2013). Hawkes model for price and trades high-frequency dynamics. Preprint. Available at . arXiv:1301.1135 · Zbl 1402.91750
[8] Barczy, M., Ispány, M. and Pap, G. (2011). Asymptotic behavior of unstable \(\mathrm{INAR}(p)\) processes. Stochastic Process. Appl. 121 583-608. · Zbl 1241.62122
[9] Bauwens, L. and Hautsch, N. (2004). Dynamic latent factor models for intensity processes. Working paper, UCL-CORE Center for Operations Research and Econometrics.
[10] Billingsley, P. (1968). Convergence of Probability Measures . Wiley, New York. · Zbl 0172.21201
[11] Bouchaud, J.-P., Gefen, Y., Potters, M. and Wyart, M. (2004). Fluctuations and response in financial markets: The subtle nature of “random” price changes. Quant. Finance 4 176-190.
[12] Bowsher, C. G. (2007). Modelling security market events in continuous time: Intensity based, multivariate point process models. J. Econometrics 141 876-912. · Zbl 1418.62375
[13] Brémaud, P. and Massoulié, L. (2001). Hawkes branching point processes without ancestors. J. Appl. Probab. 38 122-135. · Zbl 0983.60048
[14] Chavez-Demoulin, V., Davison, A. C. and McNeil, A. J. (2005). Estimating value-at-risk: A point process approach. Quant. Finance 5 227-234. · Zbl 1118.91353
[15] Cox, J. C., IngersollJr, J. E. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53 385-407. · Zbl 1274.91447
[16] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes . Springer, New York. · Zbl 0657.60069
[17] Embrechts, P., Liniger, T. and Lin, L. (2011). Multivariate Hawkes processes: An application to financial data. J. Appl. Probab. 48A 367-378. · Zbl 1242.62093
[18] Errais, E., Giesecke, K. and Goldberg, L. R. (2010). Affine point processes and portfolio credit risk. SIAM J. Financial Math. 1 642-665. · Zbl 1200.91296
[19] Filimonov, V. and Sornette, D. (2012). Quantifying reflexivity in financial markets: Toward a prediction of flash crashes. Phys. Rev. E (3) 85 056108.
[20] Filimonov, V. and Sornette, D. (2013). Apparent criticality and calibration issues in the Hawkes self-excited point process model: Application to high-frequency financial data. Preprint. Available at . arXiv:1308.6756 · Zbl 1402.91946
[21] Hardiman, S. J., Bercot, N. and Bouchaud, J.-P. (2013). Critical reflexivity in financial markets: A Hawkes process analysis. Preprint. Available at . arXiv:1302.1405
[22] Hawkes, A. G. (1971). Point spectra of some mutually exciting point processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 33 438-443. · Zbl 0238.60094
[23] Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58 83-90. · Zbl 0219.60029
[24] Hawkes, A. G. and Oakes, D. (1974). A cluster process representation of a self-exciting process. J. Appl. Probab. 11 493-503. · Zbl 0305.60021
[25] Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6 327-343. · Zbl 1384.35131
[26] Hewlett, P. (2006). Clustering of order arrivals, price impact and trade path optimisation. In Workshop on Financial Modeling with Jump Processes , 6 - 8 September 2006. Ecole Polytechnique, France.
[27] Jacod, J. (1974/75). Multivariate point processes: Predictable projection, Radon-Nikodým derivatives, representation of martingales. Z. Wahrsch. Verw. Gebiete 31 235-253. · Zbl 0302.60032
[28] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaften 288 . Springer, Berlin. · Zbl 0635.60021
[29] Jaisson, T. (2013). Market impact as anticipation of the order flow imbalance. Working paper. · Zbl 1398.91526
[30] Jaisson, T. and Rosenbaum, M. (2014). Limit theorems for nearly unstable Hawkes processes: Version with technical appendix. Technical Report 1607, Laboratoire de Probabilités et Modèles Aléatoires, Univ. Pierre et Marie Curie. · Zbl 1319.60101
[31] Jakubowski, A., Mémin, J. and Pagès, G. (1989). Convergence en loi des suites d’intégrales stochastiques sur l’espace \(\mathbf{D}^{1}\) de Skorokhod. Probab. Theory Related Fields 81 111-137. · Zbl 0638.60049
[32] Kalashnikov, V. (1997). Geometric Sums : Bounds for Rare Events with Applications : Risk Analysis , Reliability , Queueing. Mathematics and Its Applications 413 . Kluwer, Dordrecht. · Zbl 0881.60043
[33] Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035-1070. · Zbl 0742.60053
[34] Large, J. (2007). Measuring the resiliency of an electronic limit order book. Journal of Financial Markets 10 1-25.
[35] Lillo, F. and Farmer, J. D. (2004). The long memory of the efficient market. Stud. Nonlinear Dyn. Econom. 8 3. · Zbl 1081.91595
[36] Ogata, Y. (1978). The asymptotic behaviour of maximum likelihood estimators for stationary point processes. Ann. Inst. Statist. Math. 30 243-261. · Zbl 0451.62067
[37] Ogata, Y. (1983). Likelihood analysis of point processes and its applications to seismological data. Bull. Inst. Internat. Statist. 50 943-961.
[38] Petrov, V. V. (1975). Sums of Independent Random Variables . Springer, New York. · Zbl 0322.60043
[39] Reynaud-Bouret, P. and Schbath, S. (2010). Adaptive estimation for Hawkes processes; application to genome analysis. Ann. Statist. 38 2781-2822. · Zbl 1200.62135
[40] Shah, P. V. and Jana, R. K. (2013). Results on generalized Mittag-Leffler function via Laplace transform. Appl. Math. Sci. ( Ruse ) 7 567-570.
[41] Wyart, M., Bouchaud, J.-P., Kockelkoren, J., Potters, M. and Vettorazzo, M. (2008). Relation between bid-ask spread, impact and volatility in order-driven markets. Quant. Finance 8 41-57. · Zbl 1140.91414
[42] Zhu, L. (2013). Central limit theorem for nonlinear Hawkes processes. J. Appl. Probab. 50 760-771. · Zbl 1306.60015
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